Coherent optical communications technologies have increased data rates by using amplitude, phase, x-polarization, and y-polarization. Conventional non-coherent optical communications use only amplitude component of optical fields. Coherent modulations, like quadrature-amplitude modulation (QAM) or amplitude and phase-shift keying (APSK), require carrier phase estimation (CPE) for demodulating. Conventionally, feedforward CPE based on phase unwrapping has been used. For example, an Mth power method is used particularly for PSK modulation formats. The Viterbi-and-Viterbi (V&V) method uses a certain nonlinear function for amplitude normalization in conjunction with the Mth power method to improve estimation accuracy of the carrier phase.
Those blind CPE methods do not perform well for high-order QAM signal constellations, which are required to realize high data rates in next-generation optical communications. In addition, those methods have a fundamental problem of phase ambiguity, caused by the Mth power method. To recover the phase ambiguity, a phase unwrapping method is typically used. A simple phase unwrapping method can cause an additional problem known as cycle slips. A cycle slip happens with an abrupt change of phase in the CPE when a phase tracking loop in a receiver experiences a temporary loss of lock due to signal distortion, or some other disturbing factors, such as nonlinear phase noise.
Differential encoding or pilot symbol insertion can be used to reduce cycle slips. However, differential encoding has a fundamental problem of the doubling bit-error rate (BER) due to error propagation. The pilot symbol insertion also has an inevitable drawback in the reduction of the spectrum efficiency because of the undesired overhead of the pilot symbols.
With forward-error correction (FEC) codes, such as low-density parity-check (LDPC) codes, the so-called turbo principle is used to cope with various impairments in optical communications. For example, turbo equalizations can reduce linear and non-linear distortions. Turbo differential decoding has been used to compensate for the degradation of error propagation in differential encoding. Cycle slip problems have been dealt with by a turbo CPE, which uses soft-decision feedback from an FEC decoder. However, feeding back the soft-decision information from the decoder to the CPE can increase the overall latency. Instead of feeding back the soft-decision information from the FEC decoder to the CPE, the latency can be reduced by feeding back to a demodulator to compensate for cycle slips. However, it does not work well if a cycle slip probability is already high at the feedforward CPE.
The cycle slip problem becomes even more severe for high-order QAM transmissions. In high-order QAM transmissions, the in-phase of reference laser signal is referred to as I, and the quadrature signal that is shifted by 90 degrees is called Q. To generate I-Q balanced perfect QAM constellations, accurate and stable bias controls for Mach-Zender modulators are highly required. However, such a bias control is not achieved in practice especially for high-speed transceivers and high-order modulations such as 1024QAM. The bias imperfection causes a problem of angular skew, where the constellation points deviate from the ideal square-grid points according to the skew angle between I axis and Q axis.
Conventionally, one can use a Gram-Schmidt orthogonalization and k-means clustering to compensate for the skew problem. Along with the skew problem, there is still a residual phase noise after the CPE. The phase noise comes from impairment such as fiber nonlinearity and laser linewidth. The residual phase noise can degrade performance for dense high-order QAM signals. Accordingly, there is a need in the art for an approach in handling the cycle slip and the angular skew as well as the phase noise for high-order QAM transmissions in high-speed optical communications systems.